Cardfather, can you please double check this? Thanks.
···
-------------- Original message --------------
the derivation of N0 (which does rely on clt) is pretty
straightforward:
the -2sd curve is:
(er-1) x hands - 2sqrt(variance x hands)
for full pay deuces wild it is:
.00762 x hands - 2sqrt(25.83462 x hands)
this can be ploted using a spreadsheet
to find the minimum take the first derivative and solve for zero:
(er-1) - 1/sqrt(variance x hands) = 0
hands = variance/(er-1)^2
which is also the formula for N0 which is the x-intercept of the -1sd
curve
to find the loss at the minimum, plug N0 into the -2sd formula:
(er-1) x variance/(er-1)^2 - 2sqrt(variance x variance/(er-1)^2))
variance/(er-1) - 2variance/(er-1) = -variance/(er-1)
which is also the formula for the kelly bankroll
so, for the -2sd curve (2.28% of results are worse assuming clt) the
highest loss is a kelly bankroll (variance/(er-1) bets) and it occurs
at N0 (variance/(er-1)^2) hands, for full pay deuces wild 5-coin
quarters this would be $4,238 at 444,930 hands
[Non-text portions of this message have been removed]